On the accuracy of physical optics

The first-order correction to the physical optics (PO) solution for the axial radiation field from parabolic reflector antennas is found. The correction is of orderk^{-1}and is in quadrature with the PO solution. By means of the correction term the accuracy of the PO surface integral is round to be better than0.22(lambda/D)^{2}dB on axis, whereDis the diameter of the reflector. Thus, the PO solution for the directivity is extremely accurate for commun reflector sizes of several wavelengthslambda.     

On codes overZ_m(Corresp.)

The factorization of Abelian codes overZ_{m}, i.e., ideals inZ_{m},G,Ga finite Abelian group, corresponding to a factorization ofmand that of G as a product of cyclic groups are considered. Quasi-Abelian codes overZ_{m}are considered and it is shown that every quasi-Abelian code overZ_{m}is a direct sum of Abelian codes overZ_{m}. A factorization of Gray codes overZ_{m}is also considered.     

Narrow-band systems and Gaussianity

The approach to Gaussianity of the outputy(t)of a narrow-band systemh(t)is investigated. It is assumed that the inputx(t)is ana-dependent process, in the sense that the random variablesx(t)andx(t + u)are independent foru > a. WithF(y)andG(y)the distribution functions ofy(t)and of a suitable normal process, a realistic boundBon the differenceF(y) -- G(y)is determined, and it is shown thatB rightarrow 0as the bandwidthomega_oof the system tends to zero. In the special case of the shot noise process begin{equation} y(t) = sum_i h(t - t_i) end{equation} it is shown that begin{equation} mid F(y) - G(y) mid < (omega_o/lambda) frac{1}{2} end{equation} wherelambda_iis the average density of the Poisson pointst_i.     

Reflector surface deviations in large parabolic antennas

One of the important factors affecting the efficiency of parabolic reflector antennas is the degree to which the surface of the reflector deviates from the true parabolic shape. For a given reflector surface it is also important to locate the focal point of the best-fit paraboloid relative to the existing feed support structure. A simple method is presented for making these measurements, and results are given for a particular 60-ft diameter reflector. Static measurements were made with the antenna axis pointed vertically upwards, and changes were also measured as a function of elevation angle and wind. When the surface errors have a Gaussian distribution relative to the focal point, simple theory indicates that the received power relative to that from a perfect reflecting surface isP/P_{0} = exp -(2pi sigma / lambda)^{2}wheresigmais the standard deviation of the surface errors andlambdais the operating wavelength.     

The effect of potential beam energy on the performance of linear beam devices

The potential energy stored in a bunched electron beam is calculated using analytical and numerical methods. The ratio of stored potential energy to the total beam energyW_{p}/W_{0}increases rapidly with the degree of bunching and with decreasing beam filling factorb/a.W_{p}/W_{0}is proportional to the beam perveance and independent of the beam voltage, as long as relativistic effects are neglected. In a velocity-modulated linear beam-type device, such as a klystron or a traveling-wave beam tube, stored potential beam energy affects both the bunching process and the process of dc to RE energy conversion adversely. For maximum conversion efficiencyW_{p}/W_{0}should be kept as low as possible. Several ways of reducingW_{p}/W_{0}are discussed.     

Bounds for truncation error in sampling expansions of band-limited signals

Forf(t)a real-valued signal band-limited to- pi r leq omega leq pi r (0 < r < 1)and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error whenf(t)is approximated at a generic timetby an appropriate selection ofN_{1} + N_{2} + 1terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi]and thus involving samples offtaken at the integer points. Results are presented for two cases: 1) the Fourier transformF(omega)is such that|F(omega)|^{2}is integrable on[-pi, pi r](finite energy case), and 2)|F(omega)|is integrable on[-pi r, pi r]. In case 1) it is shown that the truncation error magnitude is bounded above byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereEdenotes the signal energy andgis independent ofN_{1}, N_{2}and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} right)whereMis the maximum signal amplitude andhis independent ofN_{1}, N_{2}and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values ofrdose to unity.     

On the eigenfunction expansion of electromagnetic dyadic Green's functions

A relatively simple approach is described for developing the complete eigenfunction expansion of time-harmonic electric (bar{E}) and magnetic (bar{H}) fields within exterior or interior regions containing an arbitrarily oriented electric current point source. In particular, these results yield directly the complete eigenfunction expansion of the electric and magnetic dyadic Green's functionsbarbar{G}_{e}andbarbar{G}_{m}that are associated withbar{E}andbar{H}, respectively. This expansion ofbarbar{G}_{e}andbarbar{G}_{m}contains only the solenoidal type eigenfunctions. In addition, the expansion ofbarbar{G}_{e}also contains an explicit dyadic delta function term which is required for making that expansion complete at the source point. The explicit dyadic delta function term inbarbar{G}_{e}is found readily from a simple condition governing the behavior of the eigenfunction expansion at the source point, provided one views that condition in the light of distribution theory. These general expressions for the eigenfunction expansion ofbarbar{G}_{e}andbarbar{G}_{m}reduce properly to those obtained previously for special geometries by Tai.     

Dyadic Green's functions for a coaxial line

The eigenfunction expansions of the dyadic Green's functions for a coaxial line have been derived based on the method ofbarbar{G}_{m}, whereby the irrotational vector wave functionbar{L}is not needed. A dyadic boundary condition for the discontinuity of the tangential component ofbarbar{G}_{m}has been used to facilitate the derivation of the expression for the electric dyadic Green's function.     

Complexity-based induction systems: Comparisons and convergence theorems

In 1964 the author proposed as an explication of {em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that iftilde{P}'_{M}(x)is an unnormalized form of this measure, andP(x)is any computable probability measure on strings,x, thentilde{P}'_{M}geqCP(x)whereCis a constant independent ofx. The corresponding result for the normalized form of this measure,P'_{M}, is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities ofP'_{M}are used to approximate those ofP, then the expected value of the total squared error in these conditional probabilities is bounded by-(1/2) ln C. With this error criterion, and when used as the basis of a universal gambling scheme,P'_{M}is superior to Cover's measurebast. WhenHastequiv -log_{2} P'_{M}is used to define the entropy of a rmite sequence, the equationHast(x,y)= Hast(x)+H^{ast}_{x}(y)holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation.     

Capacity theorems for the relay channel

A relay channel consists of an inputx_{l}, a relay outputy_{1}, a channel outputy, and a relay senderx_{2}(whose transmission is allowed to depend on the past symbolsy_{1}. The dependence of the received symbols upon the inputs is given byp(y,y_{1}|x_{1},x_{2}). The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)Ifyis a degraded form ofy_{1}, thenC : = : max !_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})}. 2)Ify_{1}is a degraded form ofy, thenC : = : max !_{p(x_{1})} max_{x_{2}} I(X_{1};Y|x_{2}). 3)Ifp(y,y_{1}|x_{1},x_{2})is an arbitrary relay channel with feedback from(y,y_{1})to bothx_{1} and x_{2}, thenC: = : max_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y),I ,(X_{1};Y,Y_{1}|X_{2})}. 4)For a general relay channel,C : leq : max_{p(x_{1},x_{2})} min ,{I ,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}). Superposition block Markov encoding is used to show achievability ofC, and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.     

On efficient majority logic decodable codes

A particular shortening technique is applied to majority logic decodable codes of length2^{t}. The shortening technique yields new efficient codes of lengthsn = 2^{p}, wherepis a prime, e.g., a (128,70) code withd_{maj} = 16. For moderately long code lengths (e.g.,n = 2^{11} or 2^{13}), a 20-25 percent increase in efficiency can be achieved over the best previously known majority logic decodable codes. The new technique also yields some efficient codes for lengthsn = 2^{m}wheremis a composite number, for example, a (512,316) code withd_{maj} = 32code which has 42 more information bits than the previously most efficient majority logic decodable code.     

On the Schalkwijk-Kailath coding scheme with a peak energy constraint

In a recent series of papers, [2]-[4] Schalkwijk and Kailath have proposed a block coding scheme for transmission over the additive white Gaussian noise channel with one-sided spectral densityN_{0}using a noiseless delayless feedback link. The signals have bandwidthW (W leq infty)and average powerbar{P}. They show how to communicate at ratesR < C = W log (1 + bar{P}/N_{0}W), the channel capacity, with error probabilityP_{e} = exp {-e^{2(C-R)T+o(T)}}(whereTis the coding delay), a "double exponential" decay. In their scheme the signal energy (in aT-second transmission) is a random variable with only its expectation constrained to bebar{P}T. In this paper we consider the effect of imposing a peak energy constraint on the transmitter such that whenever the Schalkwijk-Kailath scheme requires energy exceeding abar{P}T(wherea > 1is a fixed parameter) transmission stops and an error is declared. We show that the error probability is degraded to a "single exponential" formP_{e} = e^{-E(a)T+o(T)}and find the exponentE(a). In the caseW = infty , E(a) = (a - 1)^{2}/4a C. For finiteW, E(a)is given by a more complicated expression.     

The fast decoding of Reed-Solomon codes using Fermat transforms (Corresp.)

It is shown thatsqrt[8]{2}is an element of order2^{n+4}inGF(F_{n}), whereF_{n}=2^{2^{n}}+1is a Fermat prime forn=3,4. Hence it can be used to define a fast Fourier transform (FFT) of as many as2^{n+4}symbols inGF(F_{n}). Sincesqrt[8]{2}is a root of unity of order2^{n+4}inGF(F_{n}), this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the casen=2, wheresqrt{2}is inGF(F_{2})=GF(17), is given.     

Comments on "A Two-Stage Rearrangeable Broadcast Switching Network"

It is shown that a result achieved in the above paper is not applicable without constraints. The statement thatn_{2} = 13in the case ofM = 3ifsqrt{N_{1}}is, at least, 6 is not always correct. Ifsqrt{N_{1}}is a multiple of 5,n_{2}= 12, and ifsqrt{N_{1}}is a multiple of 4,n_{2} = 11.     

Harmonic analysis for a class of multiplicative processes

The harmonic analysis of certain multiplicative processes of the formg(t)X(t)is considered, wheregis a deterministic function, and the stochastic processX(t)is of the formX(t)=sum X_{n}l_{[n alpha , (n+l) alpha]}(t), where a is a positive constant and theX_{n}, n=0, pm 1,pm 2, cdotsare independent and identically distributed random variables with zero means and finite variances. In particular, we show that if g is Riemann integrable and periodic, with period incommensurate withalpha, theng(t)X(t)has an autocovariance in the Wiener sense equal to the product of the Wiener autocovariances of its factors,C_{gx} = C_{g}C_{x}. Some important cases are examined where the autocovariance of the multiplicative process exists but cannot be obtained multiplicatively.     

On the inequivalence of generalized Preparata codes

Ifmis odd andsigma /in Aut GF(2^{m})is such thatx rightarrow x^{sigma^{2}-1}is1-1, there is a[2^{m+1}-1,2^{m+l}-2m-2]nonlinear binary codeP(sigma)having minimum distance 5. All the codesP(sigma)have the same distance and weight enumerators as the usual Preparata codes (which rise asP(sigma)whenx^{sigma}=x^{2}). It is shown thatP(sigma)andP(tau)are equivalent if and only iftau=sigma^{pm 1}, andAut P(sigma)is determined.     

Linear feedback rate bounds for regressive channels (Corresp.)

This article presents new tighter upper bounds on the rate of Gaussian autoregressive channels with linear feedback. The separation between the upper and lower bounds is small. We havefrac{1}{2} ln left( 1 + rho left( 1+ sum_{k=1}^{m} alpha_{k} x^{- k} right)^{2} right) leq C_{L} leq frac{1}{2} ln left( 1+ rho left( 1+ sum_{k = 1}^{m} alpha_{k} / sqrt{1 + rho} right)^{2} right), mbox{all rho}, whererho = P/N_{0}W, alpha_{l}, cdots, alpha_{m}are regression coefficients,Pis power,Wis bandwidth,N_{0}is the one-sided innovation spectrum, andxis a root of the polynomial(X^{2} - 1)x^{2m} - rho left( x^{m} + sum^{m}_{k=1} alpha_{k} x^{m - k} right)^{2} = 0.It is conjectured that the lower bound is the feedback capacity.     

Multiple-burst error-correcting cyclic product codes (Corresp.)

LetCbe the cyclic product code ofpsingle parity check codes of relatively prime lengthsn_{1}, n_{2},cdots , n_{p} (n_{1} < n_{2} < cdots < n_{p}). It is proven thatCcan correct2^{P-2}+2^{p-3}-1bursts of lengthn_{1}, andlfloor(max{p+1, min{2^{p-s}+s-1,2^{p-s}+2^{p-s-1}}}-1)/2rfloorbursts of lengthn_{1}n_{2} cdots n_{s} (2leq s leq p-2). Forp=3this means thatCis double-burst-n_{1}-correcting. An efficient decoding algorithm is presented for this code.     

On the survival of sequence information in filters (Corresp.)

Given a binary data streamA = {a_i}_{i=o}^inftyand a filterFwhose output at timenisf_n = sum_{i=0}^{n} a_i beta^{n-i}for some complexbeta neq 0, there are at most2^{n +1)distinct values off_n. These values are the sums of the subsets of{1,beta,beta^2,cdots,beta^n}. It is shown that all2^{n+1}sums are distinct unlessbetais a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space{f_n}is2^{n+1}ifbetais transcendental, ifbeta neq pm 1is rational, and ifbetais irrational algebraic but not a unit of the type mentioned. For the exceptional values ofbeta, it appears that the size of the state space{f_n}grows only as a polynomial innifmidbetamid = 1, but as an exponentialalpha^nwith1 < alpha < 2ifmidbetamid neq 1.     

Writing on dirty paper (Corresp.)

A channel with outputY = X + S + Zis examined, The stateS sim N(0, QI)and the noiseZ sim N(0, NI)are multivariate Gaussian random variables (Iis the identity matrix.). The inputX in R^{n}satisfies the power constraint(l/n) sum_{i=1}^{n}X_{i}^{2} leq P. IfSis unknown to both transmitter and receiver then the capacity isfrac{1}{2} ln (1 + P/( N + Q))nats per channel use. However, if the stateSis known to the encoder, the capacity is shown to beC^{ast} =frac{1}{2} ln (1 + P/N), independent ofQ. This is also the capacity of a standard Gaussian channel with signal-to-noise power ratioP/N. Therefore, the stateSdoes not affect the capacity of the channel, even thoughSis unknown to the receiver. It is shown that the optimal transmitter adapts its signal to the stateSrather than attempting to cancel it.